Agreement refers to the matching of words within a sentence in terms of their number (singular and plural), and in terms of gender or person. A traditional name for the concept in English and other languages is concord.

Verb-subject agreement

Matching pronouns with verbs is straightforward enough, until you come to cases like:

Neither she nor I ?am/?is/?are inclined to go.

One or both of us ?is/?are wrong.

None of the alternatives sits comfortably in those sentences. The best way out of the problem is to remake the sentence:

Both she and I are disinclined to go.

One of us is wrong, or both of us.

With indefinite pronouns (e.g. either, neither, nobody, no-one, something), those ending in -one, -body, -thing simply take a singular verb on all occasions. But with others, a plural verb is a possibility:

Any of the books he wrote is/are worth reading.

None of their suggestions appeal(s) to us.

A singular verb in such examples singles out one item, whereas the plural suggests that the writer has the whole set in mind.

Agreement between pronouns

Everyone likes ?his/?her/?their own clothes.

In strictest grammar, the pronoun should be either his or her in such cases. But the exclusiveness of opting for one gender on the other (and the clumsiness of saying ‘his or her’) makes many people use their. Because it is gender–free, their helps to maintain the generality of the statement, and in many contexts this is preferable. Their is certainly being used in this way very often in speech, and increasingly in writing. A newspaper cartoon not so long ago had the Prime Minister saying:

Everyone has to pay their tax!

The use of their in singular agreement with indefinite pronouns is accepted as ‘standard idiom’ by the Australian Government Style Manual.

Make sure that plural nouns have plural verbs, e.g. Kogalym today is one of the few Siberian oil towns which are [not is] almost habitable. What better evidence that snobbery and elitism still hold [not holds] back ordinary British people?

The manager of a large city zoo was drafting a letter to order a pair of animals. He sat at his computer and typed the following sentence: “I would like to place an order for two mongooses, to be delivered at your earliest convenience.”
He stared at the screen, focusing on that odd word mongooses. Then he deleted the word and added another, so that the sentence now read: “I would like to place an order for two mongeese, to be delivered at your earliest convenience.”
Again he stared at the screen, this time focusing on the new word, which seemed just as odd as the original one. Finally, he deleted the whole sentence and started all over. “Everyone knows no full-stocked zoo should be without a mongoose,” he typed. “Please send us two of them.”

(The word mongoose comes from mangus, a word from the Indic language Marathi; thus the plural is more widely understood to be “mongooses”.)

While either singular or plural agreement is grammatically correct, the singular is recommended in Australian Commonwealth publications—both for consistency and to present a cohesive image in references to government bodies and activities.